The Stefan problem in nonlinear heat conduction

Summary An exact parametric solution for the planar solidification of a liquid metal occupying the infinite half-space is presented. The metal is assumed to exhibit nonlinear thermal characteristics of the Storm type. Both the idealized one phase and the full two phase problems are considered. For both problems an approximate analysis of the underlying coupled transcendental equations is presented which provides initial estimates for use in a numerical scheme. Typical numerical results are given which illustrate the monotonic nature of the solution.

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Zusammenfassung Eine genaue Lösung für die ebene Solidifizierung eines flüssigen Metalls, das in einem unendlichen Halbraum liegt, wird dargestellt. Es wird angenommen, daß das Metall nonlineare thermale Elgenschaften der Stormschen Art aufweist. Das idealisierte Einphasen-Problem wie auch das voile Zweiphasen-Problem werden betrachtet. Für beide Probleme wird eine ungefähre Analyse der zugrundeliegenden gekuppelten transzendentalen Gleichungen gegeben, die Anfangswerte zur Verwendung in einer numerischen Darstellung liefert. Es werden typische Zahlenwerte gegeben, die die Monotonie der Lösung illustrieren.

     

The nonlinear optimization problem in heat conduction

In this paper we study the existence and geometric properties of an optimal configuration to a nonlinear optimization problem in heat conduction. The quantity to be minimized is , where D is a fixed domain. A nonconstant temperature distribution is prescribed on and a volume constraint on the set where the temperature is positive is imposed. Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary.Received: 6 October 2004, Accepted: 19 October 2004, Published online: 22 December 2004     

Half-line solutions of a nonlinear heat conduction problem

We solve a half-line problem for a nonlinear diffusion equation with a given time-dependent thermal conductivity at the origin. The problem reduces to a linear Volterra integral equation, which is solvable by Picard’s process of successive approximations. We analyze some explicit examples numerically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 58–65, July, 2007.     

An inverse heat conduction problem with a nonlinear source term

In this paper, the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered. The surface heat flux history of a heated conducting body will be identified. The approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data. Some numerical examples will also be presented.     

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(_{x, t}) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x₀, t) with non-zero derivative at a single sensor location x = x₀ is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x, t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.     

Uniqueness of moving boundary for a heat conduction problem with nonlinear interface conditions

In this paper, based on the maximum principle and the unique continuation theorem, we present a uniqueness result for a moving boundary of a heat problem in a multilayer medium with nonlinear interface conditions.     

A numeric-analytic method of solving a nonlinear problem of heat conduction for a right prism

Using the example of solving a nonlinear steady-state heat-conduction problem we show the efficiency of the simultaneous application of the loading, splitting, iteration, and projection-grid methods, as well as the use of a fundamental solution and the method of boundary elements for determining the temperature fields in bodies of complicated geometrical shape.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 95–99.     

Wavelet regularization for an inverse heat conduction problem

In this paper we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The Meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero.     

The inverse coefficient problem of heat conduction for an isotropic body

We discuss a method for determining the complex of thermophysical characteristics of isotropic materials based on the solution of the two-dimensional nonstationary heat conduction problem for a layer subject to a narrow-band heating by a heat flow. We take account of the finite speed of thermal action of an annular heater.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 4–9.     

Post-processing with computable error bounds for the finite element approximation of a nonlinear heat conduction problem

Email: ain{at}mcs.le.ac.uk Email: D.Kelly{at}unsw.edu.au^* Email: I.Sloan{at}unsw.edu.au^** Email: swang{at}cs.curtin.edu.au It is shown how the finite element approximation of a nonlinearheat conduction problem may be post-processed to yield enhancedapproximations to the solution and the flux at any point inthe domain. Sharp computable bounds on the accuracy of the post-processedapproximations are derived. A criterion is identified for guidingadaptive refinements of the finite element discretization. Anumerical example is given illustrating the theoretical results.     

Regularization of backward heat conduction problem

We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.     

Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem

In this paper, the differential transform is employed to discuss the behaviors of nonlinear heat conduction problem. A hybrid method of differential transform and finite difference approach is proposed to solve the transient responses of a nonlinear heat conduction problem. Different parameters of the equation and boundary conditions are considered to verify the feasibility of the proposed method to such problems. Simulation results are illustrated and discussed in comparison with the linear case. The results show that the hybrid method can achieve good results for such problems.     

A phase-change problem for an extended heat conduction model

The aim of this paper is the analysis of a free boundary value related to a generalized heat conduction model. The qualitative analysis proposed in this paper is developed also in view of the applications of suitable computational schemes to obtain quantitative results. A basic role is played by the fundamental solution of a third-order operator, explicitly determined in a previous paper. We first discuss a method to obtain the solutions to a moving boundary value problem for a third-order operator. Next, by using the properties of the function K, the free boundary value problem is reduced to a Volterra nonlinear system for which an existence and uniqueness result is given.     

The heat conduction problem for orthotropic shells with a system of cuts

Using the Fourier integral method we have solved the heat conduction problem for an orthotropic shell of arbitrary Gaussian curvature with a system of thermally insulated cuts. In the process we have taken account of heat exchange on the lateral surfaces of the shells. We have studied the influence of the anistropy properties of the material on the distribution of the perturbed temperature field. Using the example of a system consisting of two cuts we have studied the dependence of jumps in the integral characteristics of the temperature on the relative locations of the cuts. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 50–54.     

The relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient

The main objective of this article is to analyze the RF-pair approach for the relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient. The nonlinear heat conduction equation, by means of appropriate series of operators and transformations is transformed into the classical Emden–Fowler equation.